We derive Edgeworth-type expansions for Poisson stochastic integrals, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which are based on third cumulants instead of the L3 norm term found in the literature. The use of the third cumulant results into a convergence rate faster than the classical Berry-Es...

The classical combinatorial relations between moments and cumulants of random variables are generalized into covariance-moment identities for stochastic integrals and divergence operators. This approach is based on cumulant operators defined by the Malliavin calculus in a general framework that includes Itô-Wiener and Poisson stochastic integrals as well as the Lie-Wiener path space. In particu...

For many physical systems like vehicles, acceleration can be easily measured for the respective states. However, the outputs are usually affected by stochastic noise disturbance. The mentioned systems are often sensitive to noise and structural uncertainties. Furthermore, it is very difficult to estimate the multiple integrals of the signal, acceleration to velocity and velocity to position. In...

In this paper, we consider the problem of approximate solutions of set-valued stochastic differential equations. We firstly prove an inequality of set-valued Itô integrals, which is related to classical Itô isometry, and an inequality of set-valued Lebesgue integrals. Both of the inequalities play an important role to discuss set-valued stochastic differential equations. Then we mainly state th...

A combinatorial construction of the multiple stochastic integral is developed using sequences in Clifford (geometric) algebras. In particular, sequences of Berezin integrals in an ascending chain of geometric algebras converge in mean to the iterated stochastic integral. By embedding such chains within an infinite-dimensional Clifford algebra, an infinitedimensional analogue of the Berezin inte...

Stochastic differential equations in R with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W β 2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution p...